A lot of misinformation is available on this subject - partly because it is not as trivial as it may seem at first glance, partly because there is no clear definition of what we are trying to calculate.
So lets first try to define what we are exactly trying to calculate. The formula you'll find all over the web is not 'mathematically wrong' .. to be precise, it just doesn't calculate the number you need. To avoid confusion I wont repeat that formula here, but what it calculates is the EV of a bonus, _if you only take just one bonus your entire life_. If that is the masterplan, take a bonus once and then never again, then you can use the formula you'll find on many websites. (that formula and the strategy that goes with it does not maximize the EV, it minimizes the risk of ruin by 'grinding' which is playing low bet low variance.).
If however, you are a player that values the entertainment value of gambling, and are wondering whether or not on the long term you should or rather should not take bonuses, then you need an entirely different formula.
Unfortunately, the formula to mathematically determine, ahead of time what the EV of a sequence of bonuses is is complex, and moreover it requires information (like e.g. the numerical variance of the game) that a player typically does not have access to.
That doesn't mean you cannot calculate the EV of a sequence of bonuses. In fact, its really easy to do on past data. From a player point of view, you can use a simple algorithm that will update the EV of every next bonus taking into account all the past bonuses you have already received. Allow me to skip the explenation for a second and get right down to the stuff you need.
Algorithm to calculate the running EV of a sequence of bonuses.
An example ..
So lets look at a sample. We'll simplify things and have a player lose 9 times in a row, then win 1 time. He'll deposit $50 every time, and get a $50 bonus on top of it. He spins a reel of fortune that has 10 slices. 9 win nothing and one wins 9.5 times betsize.
Also notice that if our player had decided to not take a bonus on that last deposit, but just play his own 100, he would still have an EV of 400 from past bonuses ...
Also notice what a max-cashout would do here .. a 50 bonus with 10x max cash sound reasonable ? .. in this case it would COMPLETELY eliminate the player advantage ..
In fact, in this scenario a maxcash of 10x (or 500) would be the equivalent of a WR of more than 200x. (stay away from maxcash bonuses!!)
The rule of thumb.
Playing without bonus in a sequence of bonuses.
What is the effect of making deposits where you don't claim bonuses inbetween those where you do take bonuses ? Well, the formula stays the same .. total bonus will not increase, but total stake will. In the example above, if our player had not taken 10 bonuses, but only 5, alternating a deposit with a bonus and a deposit without a bonus, then at the tenth session he would have an EV of :
EV = 5*50 bonus - 1000*0.05 = 200
And so when he cashes out on the tenth session (after 5 deposits of 50 on which he claimed bonuses and 5 of 100 without bonus), he cashes out 950 on a total deposit of 750 .. or again exactly the EV ahead when the machine is at 95% (so he didn't get lucky - he got the exact average RTP).
In other words depositing without a bonus doesn't instantly mean you have no EV from past bonuses anymore. As long as you follow the rule of thumb above - you are playing at a positive EV.
Closing thought.
It's lady luck that gives out the best bonuses. It's not the best mathematicians that win the most - its the luckiest players.
Cheers,
Enzo
So lets first try to define what we are exactly trying to calculate. The formula you'll find all over the web is not 'mathematically wrong' .. to be precise, it just doesn't calculate the number you need. To avoid confusion I wont repeat that formula here, but what it calculates is the EV of a bonus, _if you only take just one bonus your entire life_. If that is the masterplan, take a bonus once and then never again, then you can use the formula you'll find on many websites. (that formula and the strategy that goes with it does not maximize the EV, it minimizes the risk of ruin by 'grinding' which is playing low bet low variance.).
If however, you are a player that values the entertainment value of gambling, and are wondering whether or not on the long term you should or rather should not take bonuses, then you need an entirely different formula.
Unfortunately, the formula to mathematically determine, ahead of time what the EV of a sequence of bonuses is is complex, and moreover it requires information (like e.g. the numerical variance of the game) that a player typically does not have access to.
That doesn't mean you cannot calculate the EV of a sequence of bonuses. In fact, its really easy to do on past data. From a player point of view, you can use a simple algorithm that will update the EV of every next bonus taking into account all the past bonuses you have already received. Allow me to skip the explenation for a second and get right down to the stuff you need.
Algorithm to calculate the running EV of a sequence of bonuses.
Code:
- Log all your sessions. For each session note :
- B = bonus.
S = stake.
- after each session, calculate the following numbers.
TB = total of all bonus.
TS = total stake over all sessions.
- after each session you can calculate the EV of the sequence up
to that point. (assuming a 5% houseedge here).
TOTAL_RUNNING_EV = TB - TS*0.05
An example ..
So lets look at a sample. We'll simplify things and have a player lose 9 times in a row, then win 1 time. He'll deposit $50 every time, and get a $50 bonus on top of it. He spins a reel of fortune that has 10 slices. 9 win nothing and one wins 9.5 times betsize.
Code:
deposit 1 : 50D + 50B. total stake = 100.
The EV after deposit 1, using formula above : 50 - 5 = 45
deposit 2 : 50D + 50B. total stake = 100.
The EV now is 90
deposit 3,4,5,6,7,8,9 : 50D + 50B. total stake = 100 each time.
The EV now is 9*50 - 9*5 = 405
deposit 10 : 50D + 50B. total stake = 100, total win = 950
The EV now is 10*50 - 10*5 = 450
The player cashes out at this point. He deposited 500, cashes
out 950 .. exactly the EV of the bonus over his own deposit.
The total stake on the machine is now 1000, total win 950 .. i.e. 95% machine.
Also notice that if our player had decided to not take a bonus on that last deposit, but just play his own 100, he would still have an EV of 400 from past bonuses ...
Also notice what a max-cashout would do here .. a 50 bonus with 10x max cash sound reasonable ? .. in this case it would COMPLETELY eliminate the player advantage ..
In fact, in this scenario a maxcash of 10x (or 500) would be the equivalent of a WR of more than 200x. (stay away from maxcash bonuses!!)
The rule of thumb.
Code:
[COLOR="Red"][SIZE="4"] [CENTER]For as long as 5% of your totalstake is lower than your
total bonus, you are playing at positive EV.
[/CENTER]
[/SIZE][/COLOR]
Playing without bonus in a sequence of bonuses.
What is the effect of making deposits where you don't claim bonuses inbetween those where you do take bonuses ? Well, the formula stays the same .. total bonus will not increase, but total stake will. In the example above, if our player had not taken 10 bonuses, but only 5, alternating a deposit with a bonus and a deposit without a bonus, then at the tenth session he would have an EV of :
EV = 5*50 bonus - 1000*0.05 = 200
And so when he cashes out on the tenth session (after 5 deposits of 50 on which he claimed bonuses and 5 of 100 without bonus), he cashes out 950 on a total deposit of 750 .. or again exactly the EV ahead when the machine is at 95% (so he didn't get lucky - he got the exact average RTP).
In other words depositing without a bonus doesn't instantly mean you have no EV from past bonuses anymore. As long as you follow the rule of thumb above - you are playing at a positive EV.
Closing thought.
It's lady luck that gives out the best bonuses. It's not the best mathematicians that win the most - its the luckiest players.
Cheers,
Enzo